There are no guarantees in the stock market. The index fund can send you a prospectus which shows what their results have been over the past decade or so, or you can find that info on line, but "past results are not a guarantee of future performance". Returns and risk generally trade off against each other; trying for higher than average results requires accepting higher than usual risk, and you need to decide which types of investments, in what mix, balance those in a way you are comfortable with.
Reinvested dividends are exactly the same concept as compounded interest in a bank account. That is, you get the chance to earn interest on the interest, and then interest on the interest on the interest; it's a (slow) exponential growth curve, not just linear. Note that this applies to any reinvestment of gains, not just automatic reinvestment back into the same fund -- but automatic reinvestment is very convenient as a default. This is separate from increase in value due to growth in value of the companies.
Yes, you will get a yearly report with the results, including the numbers needed for your tax return. You will owe income tax on any dividends or sales of shares.
Unless the fund is inside a 401k or IRA, it's just normal property and you can sell or buy shares at any time and in any amount. Of course the advantage of investing through those special retirement accounts is advantageous tax treatment, which is why they have penalties if you use the money before retirement.
Re predicting results: Guesswork and rule of thumb and hope that past trends continue a bit longer. Really the right answer is not to try to predict precise numbers, but to make a moderately conservative guess, hope you do at least that well, and be delighted if you do better... And to understand that you can lose value, and that losses often correct themselves if you can avoid having to sell until prices have recovered.
You can, of course, compute historical results exactly, since you know how much you put in when, how much you took out when, and how much is in the account now. You can either look at how rate of return varied over time, or just compute an average rate of return; both approaches can be useful when trying to compare one fund against another... I get an approximate version of this reported by my financial management software, but mostly ignore it except for amusement and to reassure myself that things are behaving approximately as expected. (As long as I'm outperforming what I need to hit my retirement goals, I'm happy enough and unwilling to spend much more time on it... and my plans were based on fairly conservative assumptions.)
If you invest $3k, it grows at whatever rate it grows, and ten years later you have $3k+X. If you then invest another $10k, you now have $3k+X+10k, all of which grows at whatever rate the fund now grows. When you go to sell shares or fractional shares, your profit has to be calculated based on when those specific shares were purchased and how much you paid for them versus when they were sold and how much you sold them for; this is a more annoying bit of record keeping and accounting than just reporting bank account interest, but many/most brokerages and investment banks will now do that work for you and report it at the end of the year for your taxes, as I mentioned.